Mathematical Modeling seminars

Framing effects in individual decision-making have puzzled economists for decades because they are hard, if at all, to explain with rational choice theories. Why should mere changes in the description of a choice problem affect decision-making? Here, we examine the hypothesis that changes in framing cause changes in the allocation of attention to the different options – measured via eye-tracking – and give rise to changes in decision-making. We document that the framing of a sure alternative as a gain – as opposed to a loss – in a risk-taking task increases the attentional advantage of the sure option and induces a higher choice frequency of that option – a finding that is predicted by the attentional drift-diffusion model (aDDM). The model also correctly predicts other key findings such as that the increased attentional advantage of the sure option in the gain frame should also lead quicker decisions in this frame. In addition, the data reveal that increasing risk aversion at higher stake sizes may also be driven by attentional processes because the sure option receives significantly more attention – regardless of frame – at higher stakes. We also corroborate the causal impact of framing-induced changes of attention on choice with an additional experiment that manipulates attention exogenously. Finally, to study the precise mechanisms underlying the framing effect we structurally estimate an aDDM that allows for frame and option-dependent parameters. The estimation results indicate that – in addition to the direct effects of framing-induced changes in attention on choice – the gain frame also causes (i) an increase in the attentional discount of the gamble and (ii) an increased concavity of utility. Our findings suggest that the traditional explanation of framing effects in risky choice in terms of a more concave value function in the gain domain is seriously incomplete and that attentional mechanisms as hypothesized in the aDDM play a key role.

Birdsong emerges when a set of highly interconnected brain areas manage to generate a complex output. The similarities between birdsong production and human speech have positioned songbirds as unique animal models for studying learning and production of this complex motor skill. In this work, we developed a low dimensional model for a neural network in which the variables were the average activities of different neural populations within the nuclei of the song system. This neural network is active during production, perception and learning of birdsong. We performed electrophysiological experiments to record neural activity from one of these nuclei and found that the low dimensional model could reproduce the neural dynamics observed during the experiments. Also, this model could reproduce the respiratory motor patterns used to generate song. We showed that sparse activity in one of the neural nuclei could drive a more complex activity downstream in the neural network. This interdisciplinary work shows how low dimensional neural models can be a valuable tool for studying the emergence of complex motor tasks

A central question in motor physiology has been whether motor cortex activity resembles muscle activity, and if not, why not? Over fifty years, extensive observations have failed to provide a concise answer, and the topic remains much debated. To provide a different perspective, we employed a novel behavioral paradigm that affords extensive comparison between time-evolving neural and muscle activity. Single motor-cortex neurons displayed many muscle-like properties, but the structure of population activity was not muscle-like. Unlike muscle activity, neural activity was structured to avoid ’trajectory tangling’: moments where similar activity patterns led to dissimilar future patterns. Avoidance of trajectory tangling was present across tasks and species. Network models revealed a potential reason for this consistent feature: low tangling confers noise robustness. Remarkably, we were able to predict motor cortex activity from muscle activity alone, by leveraging the hypothesis that muscle-like commands are embedded in additional structure that yields low tangling. Our results argue that motor cortex embeds descending commands in additional structure that ensure low tangling, and thus noise-robustness. The dominant structure in motor cortex may thus serve not a representational function (encoding specific variables) but a computational function: ensuring that outgoing commands can be generated reliably. Our results establish the utility of an emerging approach: understanding the structure of neural activity based on properties of population geometry that flow from normative principles such as noise robustness.

Spontaneous behavior in animals and humans shows a striking amount of variability both in the spatial domain (which actions to choose) and temporal domain (when to act). Concatenating actions into sequences and behavioral plans reveals the existence of a hierarchy of timescales ranging from hundreds of milliseconds to minutes. How do multiple timescales emerge from neural circuit dynamics? How do circuits modulate temporal responses to flexibly adapt to changing demands? In this talk, we will present recent results from experiments and theory suggesting a new computational mechanism generating the temporal variability underlying naturalistic behavior. We will show how neural activity from premotor areas unfolds through temporal sequences of attractors, which predict the intention to act. These sequences naturally emerge from recurrent cortical networks, where correlated neural variability plays a crucial role in explaining the observed variability in action timing. We will then discuss how reaction times in these recurrent circuits can be accelerated or slowed down via gain modulation, induced by neuromodulation or perturbations. Finally, we will present a general mechanism producing a reservoir of multiple timescales in recurrent networks.

This conference is a free event that includes a range of activities to stimulate the cross-fertilization of ideas, including invited speaker talks, workshops, micro talks, an undergraduate research competition, a contest to discover mathematical questions in biology, and plenty of networking opportunities. Today's speakers: Cassandra Extavour, William Bialek, Amy Shyer, Ankur Saxena, Jie Liang

This conference is a free event that includes a range of activities to stimulate the cross-fertilization of ideas, including invited speaker talks, workshops, micro talks, an undergraduate research competition, a contest to discover mathematical questions in biology, and plenty of networking opportunities. Today's speakers: Nathalie Dostatni, Christopher Obara, Hernan Garcia, Aaron Dinner, David Lubensky, Jianping Fu

Many networks in the nervous system possess an abundance of inhibition, which serves to shape and stabilize neural dynamics. The neurons in such networks exhibit intricate patterns of connectivity whose structure controls the allowed patterns of neural activity. In this work, we examine inhibitory threshold-linear networks whose dynamics are constrained by an underlying directed graph. We develop a set of parameter-independent graph rules that enable us to predict features of the dynamics, such as emergent sequences and dynamic attractors, from properties of the graph. These rules provide a direct link between the structure and function of these networks, and may provide new insights into how connectivity shapes dynamics in real neural circuits.

Cortical circuits receive multiple inputs from upstream populations with non-overlapping stimulus tuning preferences. Both the feedforward and recurrent architectures of the receiving cortical layer will reflect this diverse input tuning. We study how population-wide neuronal variability propagates through a hierarchical cortical network receiving multiple, independent, tuned inputs. We present new analysis of in vivo neural data from the primate visual system showing that the number of latent variables (dimension) needed to describe population shared variability is smaller in V4 populations compared to those of its downstream visual area PFC. We successfully reproduce this dimensionality expansion from our V4 to PFC neural data using a multi-layer spiking network with structured, feedforward projections and recurrent assemblies of multiple, tuned neuron populations. We show that tuning-structured connectivity generates attractor dynamics within the recurrent PFC current, where attractor competition is reflected in the high dimensional shared variability across the population. Indeed, restricting the dimensionality analysis to activity from one attractor state recovers the low-dimensional structure inherited from each of our tuned inputs. Our model thus introduces a framework where high-dimensional cortical variability is understood as ``time-sharing’’ between distinct low-dimensional, tuning-specific circuit dynamics.

Neurons in the primary visual cortex (V1) encode the orientation and contrast of visual stimuli through changes in firing rate (Hubel and Wiesel, 1962). Their activity typically peaks at a preferred orientation and decays to zero at the orientations that are orthogonal to the preferred. This activity pattern is re-scaled by contrast but its shape is preserved, a phenomenon known as contrast invariance. Contrast-invariant selectivity is also observed at the population level in V1 (Carandini and Sengpiel, 2004). The mechanisms supporting the emergence of contrast-invariance at the population level remain unclear. How does the activity of different neurons with diverse orientation selectivity and non-linear contrast sensitivity combine to give rise to contrast-invariant population selectivity? Theoretical studies have shown that in the balance limit, the properties of single-neurons do not determine the population activity (van Vreeswijk and Sompolinsky, 1996). Instead, the synaptic dynamics (Mongillo et al., 2012) as well as the intracortical connectivity (Rosenbaum and Doiron, 2014) shape the population activity in balanced networks. We report that short-term plasticity can change the synaptic strength between neurons as a function of the presynaptic activity, which in turns modifies the population response to a stimulus. Thus, the same circuit can process a stimulus in different ways –linearly, sublinearly, supralinearly – depending on the properties of the synapses. We found that balanced networks with excitatory to excitatory short-term synaptic plasticity cannot be contrast-invariant. Instead, short-term plasticity modifies the network selectivity such that the tuning curves are narrower (broader) for increasing contrast if synapses are facilitating (depressing). Based on these results, we wondered whether balanced networks with plastic synapses (other than short-term) can support the emergence of contrast-invariant selectivity. Mathematically, we found that the only synaptic transformation that supports perfect contrast invariance in balanced networks is a power-law release of neurotransmitter as a function of the presynaptic firing rate (in the excitatory to excitatory and in the excitatory to inhibitory neurons). We validate this finding using spiking network simulations, where we report contrast-invariant tuning curves when synapses release the neurotransmitter following a power- law function of the presynaptic firing rate. In summary, we show that synaptic plasticity controls the type of non-linear network response to stimulus contrast and that it can be a potential mechanism mediating the emergence of contrast invariance in balanced networks with orientation-dependent connectivity. Our results therefore connect the physiology of individual synapses to the network level and may help understand the establishment of contrast-invariant selectivity.

The dream of programmable matter is to create materials whose physical properties (shape, moduli, response to perturbations, etc.) can be changed on the fly. For many years, my group has been thinking about how to program flat sheets that fold up into three dimensional shapes, most recently by exploiting the principles of origami design. Unfortunately, a combinatorial explosion of folding pathways makes robust folding particularly challenging. In this talk, I will discuss how this pluripotency arises from the topology of the configuration space. This suggests a broader understanding of a larger class of materials spanning from folding forms to spring networks to mechanical structures that perform computational logic.

Neural integrators are circuits that are able to code analog information such as spatial location or amplitude. Storing amplitude requires the network to have a large number of attractors. In classic models with recurrent excitation, such networks require very careful tuning to behave as integrators and are not robust to small mistuning of the recurrent weights. In this talk, I introduce a circuit with recurrent connectivity that is subjected to a slow subthreshold oscillation (such as the theta rhythm in the hippocampus). I show that such a network can robustly maintain many discrete attracting states. Furthermore, the firing rates of the neurons in these attracting states are much closer to those seen in recordings of animals. I show the mechanism for this can be explained by the instability regions of the Mathieu equation. I then extend the model in various ways and, for example, show that in a spatially distributed network, it is possible to code location and amplitude simultaneously. I show that the resulting mean field equations are equivalent to a certain discontinuous differential equation.

In a cross wind, the direction a fly moves through the air may differ from its heading direction, the direction defined by its body axis. I will present a model based on experimental results that reveals how a heading direction “compass” signal is combined with optic flow to compute and represent the direction that a fly is traveling. This provides a general framework for understand how flies perform vector computations.

The coordinated migration of groups of cells underlies many biological processes, including embryo development, wound healing and cancer metastasis. In many of these situations, tissues are able to tune themselves between liquid-like states, where cells flow collectively as in a liquid, and solid-like states that can support shear stresses. In this talk I will describe mesoscopic models of cell assemblies inspired by active matter physics to examine the roles of cell motility, cell crowding and the interplay of contractility and adhesion in controlling the rheological state of biological tissue.

When was the last time you ran into a giant? Chances are never. Almost 100 years ago, JBS Haldane posed an outwardly simple yet complex question – what is the most optimal size (for a biological system)? The living world around us contains a huge diversity of organisms, each with its own characteristic size. Even the size of subcellular organelles is tightly controlled. In absence of physical rulers, how do cells and organisms truly “know” how large is large enough? What are the mechanisms in place to enforce size control? Many of these questions have motivated generations of scientists to look for physical principles underlying size control in biological systems. In the next edition of Emory's Theory and Modeling of Living Systems (TMLS) workshop series, our panel of speakers will take a close look at these questions, across the entire scale - from the molecular, all the way to the ecosystem.

Understanding the transport of microorganisms and self-propelled particles in porous media has important consequences in human health as well as for microbial ecology. In this work, we explore models for the dispersion of active particles in both periodic and random porous media. In a first problem, we analyze the long-time transport properties in a dilute system of active Brownian particles swimming in a periodic lattice in the presence of an external flow. Using generalized Taylor dispersion theory, we calculate the mean transport velocity and dispersion dyadic and explain their dependence on flow strength, swimming activity and geometry. In a second approach, we address the case of run-and-tumble particles swimming through unstructured porous media composed of randomly distributed circular pillars. There, we show that the long-time dispersion is described by a universal hindrance function that depends on the medium porosity and ratio of the swimmer run length to the pillar size. An asymptotic expression for the hindrance function is derived in dilute media, and its extension to semi-dilute and dense media is obtained using stochastic simulations. We conclude by discussing the role of hydrodynamic interactions and swimmer concentration effects.

The ability of the brain to encode and store information depends on the plastic nature of the individual synapses. The increase and decrease in synaptic strength, mediated through the structural plasticity of the spine, are important for learning, memory, and cognitive function. Dendritic spines are small structures that contain the synapse. They come in a variety of shapes (stubby, thin, or mushroom-shaped) and a wide range of sizes that protrude from the dendrite. These spines are the regions where the postsynaptic biochemical machinery responds to the neurotransmitters. Spines are dynamic structures, changing in size, shape, and number during development and aging. While spines and synapses have inspired neuromorphic engineering, the biophysical events underlying synaptic and structural plasticity of single spines remain poorly understood. Our current focus is on understanding the biophysical events underlying structural plasticity. I will discuss recent efforts from my group — first, a systems biology approach to construct a mathematical model of biochemical signaling and actin-mediated transient spine expansion in response to calcium influx caused by NMDA receptor activation and a series of spatial models to study the role of spine geometry and organelle location within the spine for calcium and cyclic AMP signaling. Second, I will discuss how mechanics of membrane-cytoskeleton interactions can give insight into spine shape region. And I will conclude with some new efforts in using reconstructions from electron microscopy to inform computational domains. I will conclude with how geometry and mechanics plays an important role in our understanding of fundamental biological phenomena and some general ideas on bio-inspired engineering.

Clock networks containing the same central architectures may vary drastically in their potential to oscillate, raising the question of what controls robustness, one of the essential functions of an oscillator. We computationally generate an atlas of oscillators and found that, while core topologies are critical for oscillations, local structures substantially modulate the degree of robustness. Strikingly, two local structures, incoherent and coherent inputs, can modify a core topology to promote and attenuate its robustness, additively. The findings underscore the importance of local modifications to the performance of the whole network. It may explain why auxiliary structures not required for oscillations are evolutionary conserved. We also extend this computational framework to search hidden network motifs for other clock functions, such as tunability that relates to the capabilities of a clock to adjust timing to external cues. Experimentally, we developed an artificial cell system in water-in-oil microemulsions, within which we reconstitute mitotic cell cycles that can perform self-sustained oscillations for 30 to 40 cycles over multiple days. The oscillation profiles, such as period, amplitude, and shape, can be quantitatively varied with the concentrations of clock regulators, energy levels, droplet sizes, and circuit design. Such innate flexibility makes it crucial to studying clock functions of tunability and stochasticity at the single-cell level. Combined with a pressure-driven multi-channel tuning setup and long-term time-lapse fluorescence microscopy, this system enables a high-throughput exploration in multi-dimension continuous parameter space and single-cell analysis of the clock dynamics and functions. We integrate this experimental platform with mathematical modeling to elucidate the topology-function relation of biological clocks. With FRET and optogenetics, we also investigate spatiotemporal cell-cycle dynamics in both homogeneous and heterogeneous microenvironments by reconstructing subcellular compartments.

Intrinsically disordered proteins (IDPs), lack of a well-defined folded structure, have been recently shown to be critical to forming membrane-less organelles via liquid-liquid phase separation (LLPS). Due to the flexible conformations of IDPs, it could be challenging to investigate IDPs with solely experimental techniques. Computational models can therefore provide complementary views at several aspects, including the fundamental physics underlying LLPS and the sequence determinants contributing to LLPS. In this presentation, I will start with our coarse-grained computational framework that can help generate sequence dependent phase diagrams. The coarse-grained model further led to the development of a polymer model with empirical parameters to quickly predict LLPS of IDPs. At last, I will show our preliminary efforts on addressing molecular interactions within LLPS of IDPs using all-atom explicit-solvent simulations.

Humans spend a lifetime learning, storing and refining a repertoire of motor memories. However, it is unknown what principle underlies the way our continuous stream of sensori-motor experience is segmented into separate memories and how we adapt and use this growing repertoire. Here we develop a principled theory of motor learning based on the key insight that memory creation, updating, and expression are all controlled by a single computation – contextual inference. Unlike dominant theories of single-context learning, our repertoire-learning model accounts for key features of motor learning that had no unified explanation and predicts novel phenomena, which we confirm experimentally. These results suggest that contextual inference is the key principle underlying how a diverse set of experiences is reflected in motor behavior.